Answering your suggestion about the stars Beta Scorpius, Omega Scorpius, and Kappa Scorpius B. I note that Beta Scorpii is about 530 light years from Earth, Omega I Scorpii is about 423 light years from Earth, Omega 2 Scorpii is about 265 light years from Earth, and Kappa Scorpii is about 480 light years from Earth. So even if the three stars were perfectly lined up as seen from Earth, instead of many degrees of angle between them, Beta Scorpii and Kappa Scorpii would be about 50 light years apart, and either Omega Scorpii would be tens or hundreds of light years beyond. So the
Enterprise might visit tens or hundreds of other stars while visiting those three stars.
https://en.wikipedia.org/wiki/List_of_stars_in_Scorpius
The best choice for having the stars close enough for your suggestion would be a constellation in which the Omega star was in a star cluster with a lot of other bright stars very close to it.
Many but not all constellations have stars with the Greek letter Omega. Some have more than one Omega star, in those cases there is a number, supposed to be a superscript number, after the Omega.
I believe the list of the 39 Omega stars in constellations is:
1. Omega Andromedae
2. Omega1 Aquarii
3. Omega2 Aquarii.
4. Omega1 Aquilae
5. Omega2 Aquilae
6. Omega Aurigae
7. Omega Bootis
8. Omega1 Cancri
9, Omega2 Cancri
10. Omega Canis Majoris
11. Omega Capricorni
12. Omega Carinae
13. Omega Cassiopeiae
14. Omega Centauri (which is actually a globular star cluster)
15. Omega1 Cygni
16, Omega2 Cygni
17. Omega Draconis
18. Omega Eridani
19. Omega Fornacis
20. Omega Geminorum
21. Omega Herculis
22. Omega Hydrae
23. Omega Leonis
24. Omega Lupi
25. Omega Octanis
26. Omega Ophiuchi
27. Omega Orionis
28. Omega Pavonis
29. Omega Persei
30. Omega Phoenicis
31. Omega Piscium
32. Omega Sagittarii
33. Omega1 Scorpii
34. Omega2 Scorpii
35. Omega Serpentis
36. Omega1 Tauri
37. Omega2 Tauri
38. Omega Ursae Majoris
39. Omega Virginis
The modern official list of constellations selected by the International Astronomical Union in 1928 has 88 constellations. Many other constellations were proposed by various astronomers in western civilization and more or less accepted for a while but have been discontinued. It is possible that some of the discontinued constellations had stars with omega designations. Many other cultures had totally different constellations but I don't now if anyone ever gave Greek letter designations to stars in those constellations.
In the Dune series several stars have designations combining Greek letters with the names of exotic and/or fictional constellations: Delta Kaising, Theta Salish, Theta Shaowei, Gamma Waiping.
In
Star Trek many stars have been given designations similar to the ones that Bayer gave to naked eye stars, with Greek letters followed by the genitive cases of the Latin names of those constellations, but using the names of constellations which have never been used on Earth.
Some of those names were used as early as TOS and TAS:
Beta Antares in "A Piece of the Action", Gamma Canaris & Epsilon Canaris in "Metamorphosis", Beta Niobe in "All Our Yesterdays", Beta Portolan in "Operation Annihilate!", and Delta Vega in "Where No Man Has Gone Before".
My theory is that some distant Earth colony or colonies created new constellations and gave the stars "in" those new constellations Bayer-type designations. And it is possible that every one of those constellations has an Omega star.
The list of those possible new constellations in
Star Trek is:
1. Agni
2. Antares (also the name of a real star).
3. Argus
4. Arigulon
5. Ashanti
6. Ataru
7. Balder
8. Berman
9. Bowles
10. Braga
11. Canaris
12. Cassius
13. Coupsic
14. Curry
15. Eminiar. (also the name of a fictional star)
16. Erandi
17. Erani
18. Eridon
19. Hutzel
20. Lankal
21. Laputa.
22. Legato
23. Magellan
24. Mahoga
25. Mees
26. Mynos
27. Nesterowitz
28. Nel
29. Niobe
30. Onais
31. Pascal
32. Penthe
33. Portolan
34. Rana (also the name of a real star)
35. Reilley
36. Renner
37. Shiro
38. Silar
39. Simmons
40. Stromgren
41. Thoridar
42. Vega (Also the name of a real star)
43. Wiltz
http://www.ex-astris-scientia.org/inconsistencies/bayer-names.htm
The only one of those "new Constellations" that has two stars named is Canaris, and that is in the same episode. Thus one might suppose that the number of those new constellations is great enough that it is statistically improbable for stars in the same "new constellation" to be mentioned in different episodes. So it is possible that there are about 100 to 10,000 "new constellations" with possibly 100 to 10,000 stars with Omega designations.
Since the number of stars with Bayer type designations including Omega is limited by the ones in present constellations and by the number of hypothetical "new constellations" in the skies of Earth colonies, Those stars would be only a tiny minority in even the small part of the galaxy that is explored in . Thus it seems likely that the "omega" in Omega IV is part of some catalog designation which uses long,complex combinations of Latin letters, Greek letters, and Arab numbers to designate the stars, and that the complex designation of that star is simplified to "Omega" by the speakers in the episode."The Omega Glory".